2002 2002 AgilentAgilent TechnologiesTechnologiesEurophysics Prize LectureEurophysics Prize Lecture
on
Bernard Barbara, L. Néel Lab, Grenoble, FranceJonathan R. Friedman, Amherst College, Amherst, MA, USADante Gatteschi, University of Florence, ItalyRoberta Sessoli, University of Florence, ItalyWolfgang Wernsdorfer, L. Néel Lab, Grenoble, France
Budapest 26/08/2002
Quantum Dynamics of Nanomagnets
the miniaturization process
Single Domain Particles
coherent rotation of all the spins
θ
Ener
gy
θ
∆E
Quantum effects in thedynamics of the magnetizationFirst evidences of Quantum Tunneling innanosized magnetic particles
(difficulties due to size distribution)
0 3 nmQuantum Coherence in ferrihydriteconfined in the ferritin mammalianprotein
(inconclusive due to distribution of ironload)
= metal ions = oxygen = carbon
The molecules are regularly arranged in the crystal
Mn(IV)S=3/2
Mn(III)S=2
Total Spin=10
Mn12acetateMn12acetate
T. Lis Acta Cryst. 1980, B36, 2042.
high spin molecules
and low spin molecules
Uniaxial magnetic anisotropy H=-DSz
2
H=0
M
M=+S
M=-S+1M=S-1
0
E
-S+S
If S is large
H=-DSz2+gµBHzSz
H≠0
M=-S
M=S
return to the equilibriumthermal activated mechanismthermal activated mechanism
H=0
∆∆∆∆E=DS2
M=-SM=S
τ=τ0exp(∆E/kBT) τ0≈10-7
∆E=63 K
time=0
J. Villain et al. Europhys. Lett.1994, 27, 159
return to the equilibriumthermal activated mechanismthermal activated mechanism
H=0
∆∆∆∆E=DS2
M=-SM=S
τ=τ0exp(∆E/kBT) τ0≈10-7
∆E=63 K
time=∞
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(τ/τ 0
)]-1
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(τ/τ 0
)]-1 τ
Sessoli et al. Nature 1993, 365, 141
τ0=2x10-7 s
∆E/kB=61 K
Temperature dependence of the relaxation time of Mn12acetate
Mn12acetate: Mn12acetate: HysteresisHysteresis loop loop
-3 -2 -1 0 1 2 3
-20
-10
0
10
20T=2.1K
M
AG
NE
TIZA
TIO
N ( µµ µµ
B)
MAGNETIC FIELD (T)
magnetic hysteresiswithout cooperativity
High Spin Clusters
Single Molecule Magnets
applications?
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(τ/τ 0
)]-1
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(τ/τ 0
)]-1 τ
τ0=2x10-7 s
∆E/kB=61 K
Temperature dependence of the relaxation time of Mn12acetate
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(τ/τ 0
)]-1
deviations fromthe Arrhenius law
Barbara et al. J. Magn. Magn. Mat. 1995, 140-144, 1825
return to the equilibriumtunnel tunnel mechanismmechanism
H=0
M=S M=-S
terms in Sx and Sy of the spin Hamiltonian
return to the equilibriumtunnel tunnel mechanismmechanism
H=0
M=S M=-S
terms in Sx and Sy of the spin Hamiltonian
What is the difference ?
Four fold axisTetragonal (E=0)
Mn12 Fe8
HH = = µµBB S.g.BS.g.B - D - D SSzz22 + + E (E (SSxx
22--SSyy22)) + B + BSSzz
44 + + C (C (SS++44++SS--
44))
Stot=10
What is the difference ?
Four fold axisTetragonal (E=0)
Two fold axisRhombic (E≠0)
Mn12 Fe8
HH = = µµBB S.g.BS.g.B - D - D SSzz22 + + E (E (SSxx
22--SSyy22)) + B + BSSzz
44 + + C (C (SS++44++SS--
44))
Hysteresis loops for Mn12
Friedman et al.,PRL, 1996;Hernandez et al,EPL, 1996;Thomas et al.,Nature, 1996
-0.4-0.3
-0.2
-0.1
00.1
0.2
0.3
0.4
-30 -20 -10 0 10 20 30
2.0 K2.2 K2.4 K2.6 K2.8 K3.0 Κ
M (e
mu)
H (kOe)
Hysteresis loops for Mn12
−5
0
5
10
15
20
25
30
0 5 10 15 20
2.0 K2.2 K2.4 K2.6 K2.8 K3.0 K
dM/d
H (1
0-5 e
mu/
Oe)
H (kOe)
Friedman et al.,PRL, 1996;Hernandez et al,EPL, 1996;Thomas et al.,Nature, 1996
Uniform spacing betweensteps
0
5
10
15
20
25
30
0 1 2 3 4 5 6
Hef
f (kO
e)
step number n
Step spacing: ~4.5 kOe
-0.4-0.3
-0.2
-0.1
00.1
0.2
0.3
0.4
-30 -20 -10 0 10 20 30
2.0 K2.2 K2.4 K2.6 K2.8 K3.0 Κ
M (e
mu)
H (kOe)
Hysteresis loops for Mn12
Enhanced Relaxation at Step Fields
10-3
10-2
0 2000 4000 6000
9.5 kOe9.0 kOe
(Msa
t - M
) (e
mu)
t (s)
Higher energy barrier
Yet faster relaxation!
Enhanced Relaxation at Step Fields
10−5
10−4
10−3
10−2
0 5 10 15 20
2.0 Κ2.6 Κ
Γ (s-1)
H (kOe)
Thermally Assisted ResonantTunneling
m = -10
m = -9
m = 10
m = 9
Thermalactivation
Fast tunneling
Tunneling occurs when levels in opposite wells align.
Hamiltonian for Mn122z BDS gµ= − − ⋅S H
The field at which (in the left well) crosses (in the right well):
m nm+−
Bnmm g
DnHµ−=+−,
Steps occur at regular intervals of field, as observed.
Step occurs every 4.5 kOe ⇒ D/g = 0.31 K
Compare with ESR data:D = 0.56 K, g = 1.93 D/g = 0.29 K(Barra et al., PRB, 1997)
Hamiltonian for Mn122z BDS gµ= − − ⋅S H 4
zBS−Spectroscopic studies revealed a 4th-order longitudinal anisotropy term B ~1.1 mK. (ESR: Barra et al., PRB, 1997 and Hill et al., PRL, 1998; INS:Mirebeau et al., PRL, 1999, Zhong et al., JAP, 2000 and Bao et al., cond-mat, 2000)
⇒Different pairs of levels cross at slightly different fields.
⇒Allows for the Examination of the Crossover from Thermally Assisted toPure Quantum Tunneling.
Crossover to Ground-stateTunneling
Abrupt “first-order” transition betweenthermally assisted and ground statetunneling.
Theory: Chudnovsky and Garanin, PRL,1997; Exp’t: Kent, et al., EPL, 2000, Merteset al., JAP, 2001.
2 2, 1 ( )m m
B
Dn BH m mg Dµ′
′= + +
Level crossing fields:
Fe8 Hamiltonian in Zero Field
2 2 2 4 4( ) ( )z x yDS E S S C S S+ −= − + − + +
Easy Axis Hard Axis
Spin wants to rotate in the y-z plane
Two Paths for MagnetizationReversal
Easy axis
Hard axis
Z
Y
XH
ϕϕϕϕ
A
B
ClockwiseCounterclockwise
Destructive Topological InterferenceEasy axis
Hard axis
Z
Y
XH
ϕϕϕϕ
A
B
Equivalence between paths ismaintained when H is appliedalong the Hard Axis.
Topological (Berry’s) phasedepends on solid angle Ωenscribed by the two paths.
Complete destructiveinterference occurs for certaindiscrete values of Ω.
Theoretical Prediction: A. Garg., 1993.
Solid AngleΩΩΩΩ
Destructive Topological Interference
A. Garg., 1993.
Modulation of Tunnel Splitting:
where Ω depends on the field along the Hard Axis.
When SΩΩΩΩ = ππππ/2, 3ππππ/2, 5ππππ/2…, tunneling is completely suppressed!
Interval between such destructive interference points:
cos( ),S∆ = Ω
2 2 ( )B
H E E Dgµ
∆ = +
Measured Tunnel Splitting
0 0.2 0.4 0.6 0.8 1 1.2
0.1
1
10
Tunn
el s
pitti
ng ²(
10-7
K)
Magnetic tranverse field (T)
0°
ϕϕϕϕ = 90°50°
30°20°
10°
5°
0 0.2 0.4 0.6 0.8 1 1.2 1.40.1
1
10
Tunn
el s
plitt
ing
²(10
-7 K
)
Magnetic transverse field (T
M = -10 -> 10
ϕϕϕϕ - 0°
ϕϕϕϕ - 7°
ϕϕϕϕ - 20°ϕϕϕϕ - 50°ϕϕϕϕ - 90°
experimentalcalculated with
D = -0.29, E = 0.046, C = -2.9x10-5 K
W. Wernsdorfer and R. Sessoli, Science, 1999.
Parity Effect: Odd vs. EvenResonances
-1 -0.5 0 0.5 10.1
1
10
²tu
nn
el(1
0-8
K)
µ 0Htrans (T)
n = 0
n = 1
n = 2
ϕϕϕϕ - 0°
W. Wernsdorfer and R. Sessoli, Science, 1999.
What Causes Tunneling andWhy the Parity Effect in Fe8
• Tunneling is produced by terms in theHamiltonian that do not commute withSz.
• For Fe8, these terms are
• Selection rule:• Every other tunneling resonance is
forbidden!
2 2 2 2( ) ( )2x yEE S S S S+ −− = +
,...)3,2,1(2 =±=∆ ppm
What Causes Tunneling andWhy the Parity Effect in Fe8
n = 1
-9 89
10-10
n = 0
-109-910
2m p∆ = ± 2m p∆ ≠ ±Tunneling Allowed Tunneling Forbidden
Parity Effect: Odd vs. EvenResonances
-1 -0.5 0 0.5 10.1
1
10
²tu
nn
el(1
0-8
K)
µ 0Htrans (T)
n = 0
n = 1
n = 2
ϕϕϕϕ - 0°
W. Wernsdorfer and R. Sessoli, Science, 1999.
Crossover From Classical to Quantum Regime
0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,00,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
n=5
n=0
n=1
n=2
n=3n=4
n=6n=7
n=8n=9n=10
Bn
T (K)
ActivatedTunneling
Measured ( ) and Calculated ( ) Resonance Fields
Barbara et al, JMMM 140-144, 1891 (1995) and J. Phys. Jpn. 69, 383 (2000)Paulsen, et al, JMMM 140-144, 379 (1995); NATO, Appl. Sci. 301, Kluwer (1995)
Classical ThermalActivation
Tblocking
Ground-stateTunneling
Tc-o
(Mn12-ac)
The Tunnel Window:An effect of weak Hyperfine Interactions
• Chiorescu et al, PRL, 83, 947 (1999)• Barbara et al, J. Phys. Jpn. 69, 383
(2000)• Kent et al, EPL, 49, 521 (2000)
3,75 3,80 3,85 3,90 3,95 4,00 4,05 4,10 4,150
1
2
3
4
n=8T=0.95 K
dm /
dB0
B0 (T)
8-1 8-0
Inhomogeneous broadening ofTwo resonances: Dipolar fields
Data points and calculated lines
Level Scheme
0,4 0,6 0,8 1,0 1,2 1,4
3,0
3,5
4,0
4,5
5,0 10-010-1
9-09-1 9-2
8-08-1 8-2
7-07-1 7-2
6-06-1 6-2
Bn (T
)
T(K)3,0 3,5 4,0 4,5 5,0
-30
-20
-10
0
10
20
(n-p) : -S+p S-n-p
9-2 10-1
9-1 10-0
9-0
8-2
8-1
8-0
7-2
7-1
7-0
6-0
6-1
6-2
E (K)
B 0 (T)
-0.04 -0.02 0 0.02 0.04 0.06 0.0810 -7
10 -6
10 -5
ΓΓ ΓΓsq
rt(s
-1)
µ 0H(T)
M in = -0.2 M s
-0.005 0 0.0054 10 -6
6 10 -6
8 10 -610 -5
2 10 -5 t0=0s
t0=10s
t 0=5s
t0=20s
t0=40s
Homogeneousbroadening of nuclearspins: Tunnel window
• Wernsdorfer et al, PRL (1999)
Effects of Strong Hyperfine Interactions:
Tetragonal symmetry(Ho in S4)
J = L+S = 8; gJ=5/4
Dipolar interactions between Ho3+ << mT
Case of Rare-earth ions: Ho3+ in Y0.998Ho0.002LiF4
HCF-Z = -B20 O2
0 - B40 O4
0 - B44 O4
4 - B60O6
0 - B64O6
4 - gJµBJH Bl
m : acurately determined by high resolution optical spectroscopy
Sh. Gifeisman et al, Opt. Spect. (USSR) 44, 68 (1978); N.I. Agladze et al, PRL, 66, 477 (1991)
Hysteresis loop of Ho3+ ions in YLiF4
Mn12-ac
Thomas et al, Nature (1996) Giraud et al, PRL, 87, 057203-1 (2001) Friedman et al, PRL (1996), Hernandez et al, EPL (1996)
Steps at Bn = 450.n (mT) Steps at Bn = 23.n (mT)
Tunneling of Mn12-ac Molecules Tunneling of Ho3+ ion
-80 -40 0 40 80 120-1,0
-0,5
0,0
0,5
1,0
200 mK 150 mK 50 mK
M/M
S
µµµµ0Hz (mT)
-20 0 20 40 60 800
100
200
300
n=0n=3
n=1
n=-1
n=2
dH/dt > 0
1/µµ µµ 0
dm/d
Hz (1
/T)
Ho3+
-1
-0,5
0
0,5
1
-3 -2 -1 0 1 2 3
1.5K1.6K1.9K2.4K
M/M
S
BL (T)
Comparison with Mn12-ac
Role of Strong Hyperfine Interactions H = HCF-Z + A.I.J
-80 -40 0 40 80 120-1,0
-0,5
0,0
0,5
1,0
200 mK 150 mK 50 mK
M/M
S
µµµµ0Hz (mT)
-20 0 20 40 60 800
100
200
300
n=0n=3
n=1
n=-1
n=2
dH/dt > 0
1/µµ µµ 0
dm/d
Hz (
1/T)
-200 -150 -100 -50 0 50 100 150 200
-180,0
-179,5
-179,0
-178,5
E (K
)
µµµµ0Hz (mT)
-7/2
7/2
5/2
-7/2
7/2
3/25/2
3/2
-5/2 -3/2-1/21/2
-5/2 -3/2
Induce Tunneling of Electronic Moments
-1/2 1/2
Avoided Level Crossings between |Ψ−, Iz> and |Ψ+, Iz’> if DI= (Iz -Iz’ )/2 integer
Co-Tunneling of Electronic and Nuclear Spins:Electro-nuclear entanglement
Exchange-biased quantum tunnelling in adimer of Mn4 molecule
W. Wernsdorfer et al, Nature 416, 406 (2002)
V15 : The Archetype of Low spin MoleculesA Mesoscopic Spin S=1/2
Anisotropy of g-factor: ~ 0.6%Ajiro et al, J..Low. Temp.
Phys. to appear (2003)Barra et al,J. Am. Chem.
Soc. 114, 8509 (1992)
Exchange interactions:Antiferromagnetic ~ several 102K
Müller, Döring, Angew. Chem. Intl.Engl., 27, 171 (1988)
Barbara et al, cond-mat / 0205141 v1; submited to PRL.
0,0 0,2 0,4 0,6 0,8 1,00,0
0,2
0,4
0,6
0,8
1,0
M/M
S
B0(T)
αααα = 130
60 mK 0.14 T/s 0.14 mT/s
V15
M(H) : Reversible and out of equilibrium
∆∆∆∆ ∼∼∼∼ 80 mK
ener
gy
magnetic field
²
| S, -m >
| S, m-n >
1 P
1 - P
| S, -m >
| S, m-n >
Adiabatic Landau-Zener Spin Rotation
« Isolated V15 » : A two-level system « without dissipation »
M(H) = dE(H)/dH
Fast sweeping rate / Weak coupling to the cryostat
Nuclear Spin-Bath :Weak Level Broadening
Low sweeping rate / Strong coupling to the cryostat
« Non-Isolated V15 » :A two-level system « with dissipation »
0,0
0,2
0,4
0,6
0,8
1,0
-0,6 -0,3 0,0 0,3 0,60,00
0,05
0,10
0,15
T=0.1 K
B0 (T)
TS=Tph (K)
(c)
M (µ
B)
M (µ
B) T = 100 mK
0.14 T/s 0.07T/s 4.4 mT/s
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,70,0
0,2
0,4
0,6
0,8
1,0(d)
B0 (T)
Measured
Calculated
Chiorescu et al, PRL 84, 3454 (2000)
M(H): Irreversible
0,0 applied field
_hωωωω ∆∆∆∆H= ∆∆∆∆0
2+(2µµµµBB0)2
1P
1-P
|1/2,1/2> |1/2,-1/2>
|1/2,1/2>|1/2,-1/2>
∆∆∆∆0
ener
gy
LZS transition at Finite Temperature (dissipative)
Phonon-bath → bottleneck modelAbragam, Bleaney, 1970; Chiorescu et al, 1999.
Nuclear spin-bath → level broadeningStamp, Prokofiev, 1998.
V15: a Gapped Spin ½ Molecule
Dzyaloshinsky-Moriya interactions: HDM= -Σ DijSixSj
The Multi-Spin Character of the Molecule(15 spins)
+
Time Reversal Symmetry D = 0 (Kramers Theorem)
Magnetism: From Macroscopic to Single atoms
S = 10 20 10 10 10 8 10 6 10 5 10 4 10 3 10 2 10 1
clusters spinmoleculesNano-particlesNano-wiressubmicron
0 3 nm
Ho
Macroscopic QuantumTunneling of Magnetization
of Single Nanoparticles easy axis
Barium ferriteNanoparticle (10 nm)
Wernsdorfer et al. et al, PRL, 79, 4014, (1997)
Tc=0.31 K
Stoner-Wohlfarth astroid
0.2
0.4
0.6
0.8
1
1.2
0° 15° 30° 45° 60° 75° 90°
Tc( θθ θθ
)/T
c(4
5°)
angle θθθθ
Tc(θ) ∝ µ 0Ha ε1/4 cotθ 1/6 1+ cotθ 2/3( )−1
Miguel and Chudnovsky, PRB (1995)
Dissipation control of LZSMolecule spins 1/2 : Gapped
V15
Ho3+ , Mn4 pairs:Cross-spin transitions, Co-tunneling
Mn4 Fe8
Quantum DysnamicsBerry Phases
Quantum Classical crossoverQuantum Dynamics, Spin Bath
Mn12-ac
Conclusion and Perspectives Quantum Tunneling at the Mesoscopic Scale(Environmental Effects on Quantum Mechanics)
Evidence for Quantum Coherence(τφ, Rabbi oscillations, … )
Manipulations of Quantum Spins, Spins Qbits(Quantum Informations and Computers)
Tunneling of single Ho3+ ionsEntangled I-J states
Ho3+
AcknowledgementsUniv. Florence & Modena:Andrea CaneschiClaudio SangregorioLorenzo SoraceAngelo RettoriAnna FortAndrea Cornia
L. Neél Lab. CNRS GrenobleE. BonetI. Chiorescu R. Giraud L. Thomas C. Thirion R. Tiron
CCNY:Myriam SarachikYicheng Zhong
U. Barcelona:Javier TejadaJoan Manel HernandezXixiang Zhang (now Hong Kong)Elias Molins
XeroxRon Ziolo
Lehman College CUNYEugene Chudnovsky
Univ. Rio de JaneiroM. Novak
Italian INFMA. LascialfariF. BorsaR. CaciuffoG. AmorettiM. Affronte
CNRS GrenobleJ. VillainA. L. BarraC. PaulsenV. Villar A. BenoitCNRS BagneuxD. Mailly
Lehman College CUNYEugene Chudnovsky
Univ. Calif. San Diego:David HendricksonSheila AubinEvan RumbergerE. Yang
Los Alamos:Rob Robinson(now ANSTO, Australia)Tim KelleyHeinz NakotteFrans Trouw (Argonne) Wei Bao
Univ. FloridaN. Aliaga, S. Bhaduri, C.Boskovic,C. Canada, C._Sanudo, M.Soler, G. Christou
Univ. P. et M. Curie, ParisV. Marvaud, M. Verdaguer,
Univ. BielefeldH. Bögge, A. Müller
U. St. PetersburgA.M. Tkachuk
Univ. Manchester R.E.P. Winpenny
Univ. KyotoH. Ajiro, T. Goto, S. Maegawa
Univ. OkkaidoY. Furukawa
AcknowledgementsJ. F. Fernandez
A. Garg
M. Leuenberger D. Loss,
A. Mukhin
S. Miyashita
N.V. Prokof'ev
A. Zvezdin
L. J. de Jongh
J. Schweizer
P.C.E. Stamp
I. Tupitsyn
